n IYATIVES

Ihe lastpiece ofoo

ellgsawDerivatives experts are desperutely searching for a way to

measure and hedge the last frontier in derivatives -correlation risk. Without this element, pricing models do not

-,rk properly, particulady in newer products such as dffi;tYaps. By Mark Parsley

./eryone, it seems, is looking lor a theory oF

-reryrhing. A long-standing quesr in physics,the seerch [or a unifring theorv has spread torhe finrnciel rvorld.

The models that asset managers use to I

:lxnage portfblios and those used by deriva- I

.s d."l.rs to price and hedge options work I

,-.t ,,f rhe rime- hrrr sonre of rhe rime rhev

don'r work at all. Something is missing thatrvould complete the picture. I'he picture inthis cese is the term structure of interest rates

and inrerest rate volatiliry, and their relation-ship rvith loreign currency rates and equiry

PflCeS.The rocket scienrists of the financial world

have become obsessed with a phenomenonr)st of rhe tirne, but some of the time rhey

that they cannot price or hedge directly butrvhich rhev see as the frnal piece in the jigsarv

of risks they must understand and control ifrhey are to manage portfolios effectively. Thatphenomenon is correlation.

Identif,ving and quantifring correlation riskhas become the derivarive markets' holy grail.The risk itself is nothing nerv. 'When

dealerstalk about rhe term structure ofinterest rates -the relationship of rares at one point in thecurve to rates at another point - they are ralk-ing about correlation. \X/hen lurures traderswith long-dated positions worry abour futureinrerest rate movements and their relationshipwith rnargin calls and the cost of borrorving tocover them, they are lvorrving about a correla-rior-r effecr. lvlodels rvhich take correlation intoaccount are believed to be behind suddencorrections benveen futures and rrur. (fonvardrate agreement) markets.

Long before ir became a buzzword, peoplewere losing n-roney because they did norunderstand the effecrs of correlation. -lraders

ofren ran their portfblios of caps and srvap-tions together, assuming a correlation of one-a perfect correlarion berween movemenrs inrhe volatiliry of the rwo products.

Caps, portfolios ofindependent options oninterest rates, and swaptions, single options onportfolios of interest rates, are clearly relatedin price. Horvever, the relationship dependsnot only on yield curves and rhe volatilities oF

forward rates, bur also on a correlation matrixof rhe volatilities of forward rates. Any bookconraining borh caps and swaptions is a corre-lation book. Because srvaprions generally tradeat lower volatilities than caps, dealers saw aneasy arbitrage: sell cap volatiliry and buy swap-rion volatiliry. In fact, the volatiliry correlationis more complex than they believed, thevolatilities did not converge and these tradesperiodica.lly made significant enough losses tomake many dealers separate rheir portfolios.

Deeper understandingAs profit margins are squeezed, the managersof these porrfolios, realizing that this is inelfi-cient given the element oF correlation rhatdoes cxrst. are studying ways to incorporltecorrelation inro their models. In rhe rvorid ofinterest-rare products, the rheory of every-thing being sought is the model that willenable them to price all contingenr interesrrate claims using one volatilirv model.

'At present rhese kinds oF models rvouldprobably require a Cray [supercomputer]because rhey would have to iterare out thewhole rerm structure oF interesr rates, but itmight be possible," says Alex \7att.s, a directorar Sakura Global Capital. "\What has drivenrhe markets fbr the lasr half dozen years is co

incorporate more instruments into one modelor theory. The locus on correlation is not a

Fundamental change, its just that people'sbasic understanding of swaps and options is alot betrer than it was five years ago. We under-stand volariliry better; 6rst it w'as bought andsold as a commodiry; then Black-Scholes gave

us a lramework for valuing itl now we can

E.urontoney I November 1993 2)

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could ha,ue bought cheap Bund/oer spreadoptions from houses rvho believed rhat thesnv would cause them to move in lockstep.

The mosr popular correlarion product (also

second-order) is the differential swap, or diff,and it is largely rhe surge in diffvolumes rhathas prompted dealers' interest in correlation.Indeed, rvhen they talk abour correlarion risk,many dealers mean specificallv rhe risk crearedby rheir assumptions on rhe correlationberween an interest rate and an exchange raterhar is embedded in a diff.

"The diffproducr made correlation risk rhemost importanc thing rhat distinguished irFrom an inlerest-rate swap or cross-currencyswap. This is because rhe bias in the price o[rhe product is directly a consequence of rheco-variance and the mosr important inputinto the co-variance is the correlation berweensome oF rhe underlying paramerers," savs\Watt.

Diffs are easy to describe and almosr impos-sible to hedge. The srvap rvriter undertalies roreceive Libor in one currengv, usually dollars,and to pav Libor in another currency with thatpayment srream denominated in dollars. Thiscreates nvo correlation problems.

First, the dea.ler is exposed to the correla-tion benveen the rwo Libors. This can beavoided relatively easily using a pair ofinterestrate swaps, by buying bonds or using luturesOT FRAS.

Second, there is the correlation berweeninteresr rates and Fx rates. Suppose the swapwriter is paying Deurschmark Libor in dollarsand receiving dollar Libor. He is funding theDeurschmark Libor payout through theDeutschmark swap marker, so even if interestrates remain the same he is exposed to rhe riskthat the dollar will strengthen leaving himshort of Deutschmarks to pay off the dollarliabiliw. Hedging this risk means taking a viewon the correlation berween interest rates andFX rates.

Does the bank believe that a rise inDeurschmark interest rates will affecr the$/Dm rate? If so, exactly how strong is thecorrelation effect - to what extent does rhebank believe that any rise in Deutschmarkrates (and so in rhe amount of money it mustpay out to the counterparry) will be compen-sated for bv a srrengrhening of theDeutschmark against the dollar?

"Cusromers are basically getting the bankto remove the uncertainry berween the rela-tionship of interest rates and currency values,that is to guarantee that movements in Euro-pean interest rates will have no impacr on rhevalue ofthe currency. The problem rve have isthat if we take a position ro hedge against themost likely scenario, rhat is that when Euro-pean rates fall European currencies willweaken, and it doesn't happen, then rve losemoney," says Ronald Tanemura, a director atSalomon Brothers.

The correlarion problem in rhe case of diffsis particularly acute. A corporare hedgertrarionale For using one shows why. Before rhernv melrdown, a corporation with Libor-

manage volatiliry in a portfolio of options.Correlation is the next stage."

Everv long-dared option contains a correla-tion component berween interest rates and theunderlving. V4rat matters to the price of anequiqv call, For example, is not todayt spot butthe lorrvard price. The forward price is thespot plus an inrerest rate component. If anoption rvrirer believes thar there is a correla-tion benveen equiry prices and interest ratemoves - such as the apparent negative correla-tion benveen the rwo at present - then simplytracking rhe historical volatiliry of spot is anillogical hedge, implying as it does zero corre-lation.

Horvever, the present wave of interest in thesub.ject srems from a Salomon Brothers issueo[ dollar-denominated Nikkei rvarranrs in themid-r98os. Although Salomon rook a while torealize that ir had created a whole nerv class ofcurrency-protecred derivatives, Cred.it Suisse

Financial Products and Bankers Trust saw itimmediately. The quanto ivas born.

The first equiry quantos rvere second-ordercorrelarion products. That is, the correlationco-elficient (the number berween plus one andminus one that indicates how well assels are

Cha-se Manhattan's llartin Cooper lavours practical, numerical ways ofhedging correlation products.

correlated) is nor applied to the option pavourdirectly. Take the dollar-denominated Nikkeiwarrants. The exrent ro rvhich the Nikkei rises

or falls rvhen the ven strenqthens againsr thedollar only affects rhe oprion pavout indi-rectly: its effecr is correlation times exchangerate volatilirv rimes Nikliei volatilirv.

The potential effects of correlation on pric-ing (and therefore on hedging) lre more intu-itively clear lor first-order products, those inrvhich correlation directlv affects the oprionpayout. The commonest of these are spreedand outperfbrmarce options.

Diffswaps puzzleClearly, no one rvill pay much for a spread oroutperformance option on rwo assets rhevbelieve are highlv correiated. Similarly, themore confident rhe option seller thar the nvoassets are highlv and srably correlated, rhe less

hedging it rviil r,rndertake. A decrease in thecorrelation benveen the underlying assets,

such as a reduction in the volatiliry of one oF

them, may increase rhe price of the option.Differing beliefs on correlation could

clearly affect rhe market: an invesror rvhobelieved thar the ER-\{ was going to lall apart

i0 Euromonev | \ovember 1993

based sterling liabilities found itself payinginterest rates that bore no relationship to thieconomic environment in which it operated.

Treasurers saw that us rates were more obvi-ously tied to the real economy and decided rorake advantage of this by opting ro swapdollar Libor for steriing Libor in dollars. Ifthe, s recession were prolonged, dollar Libor

;uld remain low, rvhile levels of political.i-rpporr for high European interest raresiooked srrong. The political and economicfactors that drive rx and interest rate correla-tions make rhose correlations difficult topredict and subject to sudden, exrreme move-ments,

Need to quantirySo how do dealers hedge rhe risk? The simplenswer is that they don't. "Correlation risk isjsentially unhedgeable," says one rrader.'Providers of these products are taking corre-lation views, over- or under-hedging and basi-

'[y being oprimisric rhat rvhat is.redgeable today will be hedgeable in a.rple of years." They can overhedge wirh

expensive foreign exchange options, disirablebe.ause diff pricing is related to rx volariliryand is therefore option-like, or in rhe interesriate futures markets.

tVhile margins were fat, this .imprecisionJid nor matter. Banks builr prorection into theprice o[ the producr. In first-order correlationproducts, such as yield-curve options, spreadoptions, basket options, better-of-rwo-assetoptions and cross-currency caps, where corre-lation directly affects rhe'opti,cn payout, thismargin might range from 5o basis points (bp)to over roo bp. In second-order products suchas diffs, providers could proteit themselvesagainst moves in correlation from o.r to r.o for:.o bp.

Now that margins are being squeezed, deal-ers are much more concerned to quanti$, andhedge correlation risk b.cause incorrecrassumprions will cost rhem their profits.

- "\7e are gerring ro a level where these thingster. ?eople are saying, 'o.re vega *is

pposed to rranslare into a certain pe<i and itdidn't. \X.\ry nor? Oh. rhere was ln inreresr ratemove at the same time. \7e should srudv rhatrorrelation so rhar we ger a more predittable?s(L'," says one banker. "The problem withcorrelation is that it is often se.n as a second-order effect and ignored. The snag with notquandfi/ing it is the same as if you ignoredSamma in options rrading. Over rime you cxn6nd yourselFwirh a probiem rhar has t".o-.E rst-ord.r- "

order effect, the currencv', is difficult. Assum-ing vou've got rhe inreresr-rare hedge lockedup, you have ro focus on the correlation riskbecause it is the orher risk you have. Now, nomatter what rhe correlarion benveen rx andinreresr rates, rhe impact upon your p&L isgoing to be minor compared to what willhappen iF you don'r hedge the interesr raterisk. But rhose variances can move around andwipe out your profit margin."

More than rhar, traders are nor comfortablewith risks they do not understand. It is fine tohave exposures, even outrighr exposures but itis not acceptable ro have an exposure that isnot understood. As yer, there is more mathe-matical musing on the correlarion problemthan practical solurions to it. There is littleconsensus on wherher correlation is stableenough a paramerer to be used in pricingmodels. There is hardly more agreemenr onhow it should be measured: one side arguesthat the apparenr instabilirv of corre.lation isdue to measuremenr inadequacies; the othercounters rhar the instabilirv is perfectly clearFrom the hisrorical dara.

Much of the work being done at themoment falls broadly into rhe category ofmeasurement. .Some people concen_trate onimprovemenrs in measuring historical correla-tion, others wreJde wirh rhe embryonicscience of stripping implied correlation fromexisting options. Improvements in measure-menr techniques for hisrorical correlation areconcentrated on garhering more data andmeasuring correlarion over shorr periods of

Implied correlation is more difficult ro find,but is the more valuable prize, because it holdsthe key to hedging correlation risk itself.

Correlation is fundamentally difficult tohedge because there is no marker in which itcan be boughr and sold directly - in much thesame way that in the early days of the optionsmarket there was no real gamma market.Option sellers could delta hedge their primarymarket risls, bur were unable to do muchabout second-order convexiry effects. Thesedays, with liquid options markets a1l priced offthe same assumptions, gamma trading is run-of-the-mil1.

Dealers foresee the crearion of such marketsfor correlation risk - but nor yer. For example,in the same way rhat an FRA is the basiccomponenr of a srvap, so a diff pne. would bethe basic componenr of the differential swap

and cross-currency swaprion. However, in rhenear Future the focus is on pricing rhe correla-tion component wirhin existing products.

'J(hile academics publish theorerical solu-tions to the diff swap problem, the six ro 12producr leaders are sharpening up rhe pracri-cal side, where possible isolating rhe market'svaluation ofcorrelation. In this way they hopeto pur rogerher an overail picture ofwhere themarket prices the correlation benveen as manymarkets and as many intra-market variables as

possible.Not only is this necessary if dealers are to

quanti8/ their own portfoliot correlation posi-tion, ir is also the only way in which they canuse_exisring products ro hedge rhar exposure.

"One way to hedge it is obviously ro createother products thar have offserring correla-t.ions. You need to have the sysrems in place tomonitor the risk so rhar you know when tohedge, what transaction ,o .r.rr. ancl whatexact characreristics that instrument shouldhave ro flarten our your risk," says one hedqer.

Stripping implied correlation from fiisr-order producrs is reasonably intuirive. Volatil-iry is a traded commodiry and so implied andhistorica.l volatilities are generally viry close.Having a good idea about rhe volatiliry levelsused by the various providers of, say, spreadoptions means rhar by comparing rhe prices ofidenticai instrumenrs their views on correla-tions can be guessrimated and valued.

Ifa house believes rhar rhe volatilities oftheunderlying assers are more closely correlatedthan recenr historical spread volariliry impliesand than the implied correlation of ih.ircomperirors' products suggesrs, then rhey willprice a spread oprion below rhose levels.

Pricing off the streetThe problem with applying the numbersgenerated in this way is that it is not clear thatmany market parricipanrs scientifically pr.icecorrelation inro their products. Impliedvolatiliry closely follows historical livelsbecause volariliry is so firndamental an inputinto options pricing models. Correlation isnot as important - indeed spread options arepriced by the Black-Scholes model wirhoutreference to correlation because the spreaditself is used as the underlying asser. Mulri-factor models do take coirelation intoaccounr, bur getring meaningful impliedcorrelation numbers involves mor. ihr.tsimply ca.lling rhe difference in rwo houses'pricing of a spread option the correlationcomponent.

The division of producrs into firsc- andsecond-order encourages this confusion andcan be unhelpful. Diffs are second-order'orrelation products, but that does not mean,rat the correlarion element is trivial. "lf a

customer wanrs ro go long European interestrates and diff rhem back inro dollars, then Iam short rhe market, I've gor ro buy bonds,"says one diff writer. "Obviously buying thebonds is much more imporrant than -i.,rg-tng rhe Fx risk, but it is simple. The second-

"The problem withcoffelation is thatit is often seen as Lsecond-order effectand ignored"

Euromoney I November 1993 J1

Aiso, the small number of banks able rooffer correlation-related products means thateffort devoted to srripping out implied corre-lations ohen resuks simply in confirming thateveryone is using rhe same practical mix oFhistorical correlaiions plus a [irrle exrra forpossible deviarions.

. "People are generally practical. They look athow everyone else is pricing and unless theythink somerhing is drasricrlly our, rhey willprice in the same ballpark," says one deriva-ilves exPert.

DBRIVATTYES

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That said, the number of players in thismarket is so small that any head start in rhisprocess, even ii: ir means using the competi-tion's products to hedge, is vital. So sensitiveare major plavers about ways of laying offcorrelation risk that many will nor discuss

these kinds oI strategies.

Proxy hedging"In the diff swap there are other instrumentsin the market rvhich have similar embeddedcorrelations rvhich have some liquidiry whichyou can go in and our ofto match your book.Ifyou have good systems and can quanriry thecorrelation risk, vou can hedge it and flattenout your book. But just what they are idrather not sa)'," says the derivarives chief at alarge us instirution. "It's not even clear thatthe people rvho are trading those productsunderstand that rhey have those correlationsembedded in."

For example, the rx/interesr rate correlationin a diff is an economic variable. The correla-tion element might be hedgeable using anequiry quanto in which, again, an economiccorrelation is embedded. So if German rates

fall, the Deutschmark will weaken but the oaxwill rise: the bank will pay less out on the diff;however, in some related ratio it will have

fewer Deutschmarks to convert into dollars topay out under the swap. But again, in somerelated fashion, this shortfall will be compen-sated for by a rise in value of the equiryquanro. Ar least thatt the idea.

Equally, path-dependent options contain a

series of correlarion assumprions that under-pins the way in which the house writing andhedging them beiieves the path of the under-lying will develop. At presenr, when looking atpath-dependent options most people use

binomial methods or Monte Carlo simula-tions (a numerical approximation) to generatedistributions. This is increasingly seen as toosimplistic. The search is on for a way of re-modelling these options to take correlationinto account and initial attemPts showmarked differences in price.

Correlation-dependent instruments inc-lude amortizing rate swaps, in which the buyerreceives a high fixed-rate payment in exchangefor taking the risk that the swap principal willamortize according to an index that representseither real or synthetic interest rates. Inessence the buyer has granted the seller a path-dependent interest rate option that containscorrelation assumptions that can be crysta1-

lized to some exrent by breaking the optioninro a bundle of caps and swaprions.

It is true that this kind of prory hedgingwas used in the eariy days of other optionsmarkets. However, the problem with tradingcomplex correlarion-related products such as

diffs to hedge correiation risks dynamically isthe lack of consistent rwo-way flow in eitherthe insrruments or the views they represent atany time. Afthough rraders claim to see trad-ing in dills and cross-currency caps, theyadmit that clienrs tend all to rvant to make thesame bet at the same time. "You can use fancy

we lose money"

models like Garch to work out probable distri-butions and look at implied correlations butnone of that is much good when ail the tradeis one-way," says one derivatives chief

In rhe longer term the search is for correla-rion outside the lew illiquid products offeredby the elite. Martin Cooper, head of TieasuryNew Products at Chase Manhattan in Londonand a champion of practical, numerical meth-ods of hedging correlation products, believes

that one of the only liquid products fromwhich you can strip our implied correlationwith some degree oF accuracy is foreignexchange options.

"You can think ofan Fx option as being onrwo assets," says Cooper. "lf you know thevolatilities oF options on I/Dm, cable [9I]and $/Dm, then by plugging the volatilitiesinto a pricing model you can imply the corre-lations." The lower the volatiliry of the cross

relative to the volatiliry of the individualcomponents, the more highly correlated therwo exchange rates,

Cooper believes that if you accePt this way

of looking at Fx options, you can take posi-tions in implied correlation by buying andselling volatiliry to remain vega neutral. Vega

measures the change in an option's pricecaused by changes in volatiliry. A vega neutralposition, one that is indifferent to smallchanges in volatiliry Cooper says, can be seen

as a correlation position.Being able to work out implied correlations

from FX options enables banks to develop a

systematic way to substitute exactly one

oprion position for another - for example, rohedge an option on an illiquid cross in moreliquid markets.

No bookrunners yetHowevet correlation trading in this way is

more theory than practice. In theory, sellers ofcorrelation-dependent options make money ifmarkets are more correlated than is implied inthe options in the same rvay that they willmake more money if they sell an option with a

higher implied volatiliry than actual. In prac-tice, the science ofpricing options rvith preciseimplied correlation coeficients is embryonic.Even if ir is being done on a transaction byrransaction basis, there is no model as yet lormonitoring correlation on a portfolio basis.

"No one rhat I know of can run a portfolioof different correlation risks together. Thar is

just theory. In terms oF practical experience

"If we hedgeagainst the mostlikely scenario andit doesn't happ€tr,

AIex Wrtt: "We understrnd volatilin* bctter.

no-onc can pur together a book o[ thcsedifferent products rnd hir i button lnd sa"'

this is my corre iarion risk benveen tl-ris

cLrrrency and this intercst rare and so r>n. Tha.doesn'r exist ar the moment. It rnight exist intz months, but ir doesn't exist :rr rhe momcnt.The best you could do is mavbe hedge corre-lations at particuiar maturin' buckets, as vourvould a clsh or srvaps book." says one hedqergrappling rvith correlation problems in hiscap and srvaption porrlblios.

Nleanwhile. as far ,rs the clients are

concerned, correlarion is oF Iimited direcrconsequence. Thev do nor generally rvorn-about correlations benveen their fbreiqnexchange positions bevonci basic assumptionson verv closely correlated currencies such as

Dm/Dfl. Thev certainly do not look at theirexposurc to correllrion, benveen inrerest rates

in one currencv and e-xchange rates benveenrhar currencv rnd anorher.

From their point of r.ierv, a better uncie r-standing of correiations on the part oi theirdellers will meln finer prices. more consis-tency of pricing benveen reiated products -such as caps and srvaptions, and a greater vari-ery of risk management products. $7hen theunificd theorv of conringenc inreresr rlreclaims is finallv cracked. ii it is, thev rviLl

beneiit From a better understanding of rhe

way interest rates work. No one is predictinewhen thar mighr happen. I

J2 Euromonev I November 1993